We have fabricated a Cooper-pair transistor (CPT) with parameters such that for appropriate voltage biases, it behaves essentially like a single Cooper-pair box (SCB). The effective capacitance of a SCB can be defined as the derivative of the induced charge with respect to gate voltage and has two parts, the geometric capacitance, C-geom, and the quantum capacitance C-Q. The latter is due to the level anticrossing caused by the Josephson coupling and is dual to the Josephson inductance. It depends parametrically on the gate voltage and its magnitude may be substantially larger than C-geom. We have detected C-Q in our CPT, by measuring the in phase and quadrature rf signal reflected from a resonant circuit in which the CPT is embedded. C-Q can be used as the basis of a charge qubit readout by placing a Cooper-pair box in such a resonant circuit.